Infinite-dimensional Vector Space Research Articles

Orbits of Maximal Vector Spaces

Let V ∞ be a standard computable infinite-dimensional vector space over the field of rationals. The lattice $$ \mathfrak{L} $$ (V ∞ ) of computably enumerable vector subspaces of V ∞ and its quotient lattice modulo finite dimension, $$ \mathfrak{L} $$ *(V ∞ ), have been studied extensively. At the same time, many important questions still remain open. In 1998, R. Downey and J. Remmel posed the question of finding meaningful orbits in $$ \mathfrak{L} $$ *(V ∞ ) [4, Question 5.8]. This question is important and difficult and its answer depends on significant progress in the structure theory for the lattice $$ \mathfrak{L} $$ *(V ∞ ), and also on a better understanding of its automorphisms. Here we give a necessary and sufficient condition for quasimaximal (hence maximal) vector spaces with extendable bases to be in the same orbit of $$ \mathfrak{L} $$ *(V ∞ ). More specifically, we consider two vector spaces, V 1 and V 2 , which are spanned by two quasimaximal subsets of, possibly different, computable bases of V ∞ . We give a necessary and sufficient condition for the principal filters determined by V 1 and V 2 in $$ \mathfrak{L} $$ *(V ∞ ) to be isomorphic. We also specify a necessary and sufficient condition for the existence of an automorphism Φ of $$ \mathfrak{L} $$ *(V ∞ ) such that Φ maps the equivalence class of V 1 to the equivalence class of V 2 . Our results are expressed using m-degrees of relevant sets of vectors. This study parallels the study of orbits of quasimaximal sets in the lattice e of computably enumerable sets, as well as in its quotient lattice modulo finite sets, e * , carried out by R. Soare in [13]. However, our conclusions and proof machinery are quite different from Soare’s. In particular, we establish that the structure of the principal filter determined by a quasimaximal vector space in $$ \mathfrak{L} $$ *(V ∞ ) is generally much more complicated than the one of a principal filter determined by a quasimaximal set in e* . We also state that, unlike in e* , having isomorphic principal filters in $$ \mathfrak{L} $$ *(V ∞ ) is merely a necessary condition for the equivalence classes of two quasimaximal vector spaces to be in the same orbit of $$ \mathfrak{L} $$ *(V ∞ ).

Interpreting Quantum Theories

Laura Ruetsche, Interpreting Quantum Theories. Oxford: Oxford University Press. Pp. xvii, 377. Most philosophical work on quantum physics has concerned simple systems. And for good reason. Even one or two particle systems can exhibit the striking features we have come to associate with quantum physics—features such as entanglement, interference, and non-locality. But as far as physics goes, such systems barely scrape the surface. In particular, they have only a finite number of degrees of freedom, which means their states are characterized by a finite number of independent parameters. The systems studied in fields such as high energy particle physics and many-body physics, meanwhile, have uncountably many degrees of freedom. They are far more complicated than the usual philosophical fare. This is untroubling as long as the conceptual heart of quantum physics can be effectively condensed down. But what if philosophers’ toy systems are fundamentally different from realistic ones? That would be a problem. And indeed, as Laura Ruetsche convincingly argues in Interpreting Quantum Theories, systems with uncountable degrees of freedom—what Ruetsche collectively calls QM ∞ —are fundamentally different from the systems that have become the stock and trade of philosophy of physics, in ways that change the interpretational project. Ruetsche’s book does two things exceedingly well. The first is to provides a (comparatively) accessible and philosophically-oriented introduction to the algebraic approach to quantum physics, which is essential for understanding QM ∞ . This is a valuable contribution. The algebraic approach is the setting for much recent philosophical work. And yet learning it is notoriously difficult, as even pedagogical texts are aimed at research-level mathematicians. Ruetsche’s book is technically demanding, but it works to bring non-mathematicians along. It is, without a doubt, the best place to enter this literature. Second, the book sets the agenda for future work on QM ∞ , by effectively and judiciously drawing important foundational questions out of a sprawling and difficult physics literature. In this regard, it is a close cousin to John Earman’s excellent Bangs, Crunches, Whimpers, and Shrieks, which has had a significant influence on subsequent philosophy of spacetime physics. 1 Philosophers of physics will be addressing the questions Ruetsche has raised, in many cases for the first time, for years to come. Interpreting Quantum Theories centers on what might be called the problem of inequivalent representations. The quantum theories that philosophers are accustomed to are set in a mathematical structure known as a (separable) Hilbert space, which is an at-most countably infinite dimensional complex vector space. Rays in a Hilbert space represent possible states of a physical system, and self- adjoint operators on that Hilbert space correspond to properties of the system. 1 John Earman, Bangs, Crunches, Whimpers, and Shrieks (Oxford: Oxford University Press, 1995).

Localization and computation in an approximation of eigenvalues

In this note we consider the problem of localization and approximation of eigenvalues of operators on infinite dimensional Banach and Hilbert spaces. This problem has been studied for operators of finite rank but it is seldom investigated in the infinite dimensional case. The eigenvalues of an operator (between infinite dimensional vector spaces) can be positioned in different parts of the spectrum of the operator, even it is not necessary to be isolated points in the spectrum. Also, an isolated point in the spectrum is not necessary an eigenvalue. One method that we can apply is using Weyl?s theorem for an operator, which asserts that every point outside the Weyl spectrum is an isolated eigenvalue.

On the Hirsch–Plotkin radical of stability groups

We study the stability group of subspace series of infinite dimensional vector spaces. In [1], the authors proved that when the vector space has countable dimension then the Hirsch–Plotkin radical of the stability group coincides with the set of all space automorphisms that fix a finite subseries and they conjectured that this would hold in all dimensions. We give a counter example of dimension 2ℵ0. We however show that in general the result remains true if the Hirsch–Plotkin radical is replaced by the Fitting group, the product of all the normal nilpotent subgroups of the stability group. We also show that the Hirsch–Plotkin radical has a certain strong local nilpotence property.

Lineability of non-differentiable Pettis primitives

Let \(X\) be an infinite-dimensional Banach space. In 1995, settling a long outstanding problem of Pettis, Dilworth and Girardi constructed an \(X\)-valued Pettis integrable function on \([0,1]\) whose primitive is nowhere weakly differentiable. Using their technique and some new ideas we show that \(\mathbf{ND}\), the set of strongly measurable Pettis integrable functions with nowhere weakly differentiable primitives, is lineable, i.e., there is an infinite dimensional vector space whose nonzero vectors belong to \(\mathbf{ND}\).

Green’s relations, regularity and abundancy for semigroups of quasi-onto transformations

Let \(V\) be an infinite-dimensional vector space and for every infinite cardinal \(n\) such that \(n\le \dim V\), let \(AE(V,n)\) denote the semigroup of all linear transformations of \(V\) whose defect is less than \(n\). In 2009, Mendes-Goncalves and Sullivan studied the ideal structure of \(AE(V,n)\). Here, we consider a similarly-defined semigroup \(AE(X,q)\) of transformations defined on an infinite set \(X\). Quite surprisingly, the results obtained for sets differ substantially from the results obtained in the linear setting.

Transformations of measures via their generalized densities

In this note we describe algorithms for obtaining formulae for transformations of measures on infinite dimensional topological vector spaces or manifolds, generated by transformations of the domains of the measures and by transformations of the range.

On some relationships between biorthogonal systems, linear topologies and the weak‐AFPP

In this paper, we obtain new results for the weak‐AFPP in abstract spaces by exploiting biorthogonal systems techniques. Firstly, we investigate the strong‐AFPP on countably infinite dimensional Hausdorff locally convex spaces. Spaces of this class are shown to be sequentially complete iff they have the hereditary FPP for totally bounded, closed convex sets. This might open a research line for the analysis of weak‐AFPP in such frames. In connection, we provide a simple criterion for the containement of ℓ1‐sequences in terms of strongly‐equicontinuous biorthogonal systems. We then establish a few results concerning the existence of Hausdorff finer vector topologies on abstract spaces having as prescribed condition the existence of such systems. The proofs are based on methods of Peck and Porta concerning building of finer vector topologies, and a classical construction of Singer which allows us to prove under rather natural conditions the existence of equicontinuous biorthogonal systems in metrizable locally convex spaces. These results are compatible with the failure of the weak‐AFPP. We also study the inverse problem by proving that every infinite dimensional vector space admits a (non‐locally convex) Hausdorff vector topology which is complete, non‐metrizable and is compatible with a bounded Hamel Schauder basis. It is shown further that such a topology has the ‐AFPP, where is the linear span of coefficient functionals associated to a Hamel basis. Finally, inspired by a result of Shapiro, we observe that if X is a non‐locally convex F‐space with an absolute basis, then the weak‐AFPP is equivalent to the fact that every bounded convex subset of X is compact.

Extra structure and the universal construction for the Witten-Reshetikhin-Turaev TQFT

A TQFT is a functor from a cobordism category to the category of vector spaces satisfying certain properties. An important property is that the vector spaces should be finite dimensional. For the WRT TQFT, the relevant 2 + 1 2+1 -cobordism category is built from manifolds which are equipped with an extra structure such as a p 1 p_1 -structure or an extended manifold structure. We perform the universal construction of Blanchet, Habegger, Masbaum, and Vogel (1992) on a cobordism category without this extra structure and show that the resulting quantization functor assigns an infinite dimensional vector space to the torus.

Infinite dimensional proper subspaces of computable vector spaces

This article examines and distinguishes different techniques for coding incomputable information into infinite dimensional proper subspaces of a computable vector space, and is divided into two main parts. In the first part we describe different methods for coding into infinite dimensional subspaces. More specifically, we construct several computable infinite dimensional vector spaces each of which satisfies one of the following:(1)Every infinite/coinfinite dimensional subspace computes Turing's Halting Set ∅′;(2)Every infinite/cofinite dimensional proper subspace computes Turing's Halting Set ∅′;(3)There exists x∈V such that every infinite dimensional proper subspace not containing x computes Turing's Halting Set ∅′;(4)Every infinite dimensional proper subspace computes Turing's Halting Set ∅′. Vector space (4) generalizes vector spaces (1) and (2), and its construction is more complicated. The same simple and natural technique is used to construct vector spaces (1)–(3). Finally, we examine the reverse mathematical implications of our constructions (1)–(4).In the second part we examine the limitations of our simple and natural method for coding into infinite dimensional subspaces described in the previous paragraph. In particular, we prove that our simple and natural coding technique cannot produce a vector space of type (4) above, and that any vector space of type (4) must have “densely many” (from a certain point of view) finite dimensional computable subspaces. In other words, the construction of a vector space of type (4) is necessarily more complicated than the construction of vector spaces of types (1)–(3). We also introduce a new statement (in second order arithmetic) about the existence of infinite dimensional proper subspaces in a restricted class of vector spaces related to (1)–(3) above and show that it is implied by weak König's lemma in the context of reverse mathematics. In the context of reverse mathematics this gives rise to two statements from effective algebra about the existence of infinite dimensional proper subspaces (for a certain class of vector spaces) of the form (∀V)[X(V)→A(V)] and (∀V)[X(V)→B(V)], that each imply ACA0 over RCA0, but such that the seemingly weaker statement (∀V)[X(V)→A(V)∨B(V)] is provable via WKL0 over RCA0. Furthermore, we highlight some general similarities between constructing of infinite dimensional proper subspaces of computable vector spaces and constructing solutions to computable instances of various combinatorial principles such as Ramsey's Theorem for pairs.

Exact solutions of one-dimensional nonlinear shallow water equations over even and sloping bottoms

We establish an equivalence of two systems of equations of one-dimensional shallow water models describing the propagation of surface waves over even and sloping bottoms. For each of these systems, we obtain formulas for the general form of their nondegenerate solutions, which are expressible in terms of solutions of the Darboux equation. The invariant solutions of the Darboux equation that we find are simplest representatives of its essentially different exact solutions (those not related by invertible point transformations). They depend on 21 arbitrary real constants; after “proliferation” formulas derived by methods of group theory analysis are applied, they generate a 27-parameter family of essentially different exact solutions. Subsequently using the derived infinitesimal “proliferation” formulas for the solutions in this family generates a denumerable set of exact solutions, whose linear span constitutes an infinite-dimensional vector space of solutions of the Darboux equation. This vector space of solutions of the Darboux equation and the general formulas for nondegenerate solutions of systems of shallow water equations with even and sloping bottoms give an infinite set of their solutions. The “proliferation” formulas for these systems determine their additional nondegenerate solutions. We also find all degenerate solutions of these systems and thus construct a database of an infinite set of exact solutions of systems of equations of the one-dimensional nonlinear shallow water model with even and sloping bottoms.

ON SOME SOLUTIONS OF A FUNCTIONAL EQUATION RELATED TO THE PARTIAL SUMS OF THE RIEMANN ZETA FUNCTION

In this paper, we prove that infinite-dimensional vector spaces of -dense curves are generated by means of the functional equations f(x)+f(2x)+<TEX>${\cdots}$</TEX>+f(nx) = 0, with <TEX>$n{\geq}2$</TEX>, which are related to the partial sums of the Riemann zeta function. These curves <TEX>${\alpha}$</TEX>-densify a large class of compact sets of the plane for arbitrary small <TEX>${\alpha}$</TEX>, extending the known result that this holds for the cases n = 2, 3. Finally, we prove the existence of a family of solutions of such functional equation which has the property of quadrature in the compact that densifies, that is, the product of the length of the curve by the <TEX>$n^{th}$</TEX> power of the density approaches the Jordan content of the compact set which the curve densifies.

Hofer geometry and cotangent fibers

For a class of Riemannian manifolds that include products of arbitrary compact manifolds with manifolds of nonpositive sectional curvature on the one hand, or with certain positivecurvature examples such as spheres of dimension at least 3 and compact semisimple Lie groups on the other, we show that the Hamiltonian diffeomorphism group of the cotangent bundle contains as subgroups infinite-dimensional normed vector spaces that are bi-Lipschitz embedded with respect to Hofer’s metric; moreover these subgroups can be taken to consist of diffeomorphisms supported in an arbitrary neighborhood of the zero section. In fact, the orbit of a fiber of the cotangent bundle with respect to any of these subgroups is quasi-isometrically embedded with respect to the induced Hofer metric on the orbit of the fiber under the whole group. The diffeomorphisms in these subgroups are obtained from reparametrizations of the geodesic flow. Our proofs involve a study of the Hamiltonian-perturbed Floer complex of a pair of cotangent fibers (or, more generally, of a conormal bundle together with a cotangent fiber). Although the homology of this complex vanishes, an analysis of its boundary depth yields the lower bounds on the Lagrangian Hofer metric required for our main results.

Classification of finite potent endomorphisms

The aim of this work is to offer a family of invariants that allows us to classify finite potent endomorphisms on arbitrary vector spaces, generalizing the classification of endomorphisms on finite-dimensional vector spaces. As a particular case we classify nilpotent endomorphisms on infinite-dimensional vector spaces.

Separation problem for a family of Borel and Baire G-powers of shift measures on $ \mathbb{R} $

The separation problem for a family of Borel and Baire G-powers of shift measures on $ \mathbb{R} $ is studied for an arbitrary infinite additive group G by using the technique developed in [L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York (1974)], [ A. N. Shiryaev, Probability [in Russian], Nauka, Moscow (1980)], and [G. R. Pantsulaia, Invariant and Quasiinvariant Measures in Infinite-Dimensional Topological Vector Spaces, Nova Sci., New York, 2007]. It is proved that $ {T_n}:{{\mathbb{R}}^n}\to \mathbb{R},\;n\in \mathbb{N} $ , defined by $$ {T_n}\left( {{x_1},\ldots,{x_n}} \right)=-{F^{-1 }}\left( {{n^{-1 }}\#\left( {\left\{ {{x_1},\ldots,{x_n}} \right\}\cap \left( {-\infty; 0} \right]} \right)} \right) $$ for (x 1,…, x n ) ∈ $ {{\mathbb{R}}^n} $ is a consistent estimator of a useful signal θ in the one-dimensional linear stochastic model $$ {\xi_k}=\theta +{\varDelta_k},\quad k\in \mathbb{N}, $$ where #(∙) is a counting measure, ∆ k , k ∈ $ \mathbb{N} $ , is a sequence of independent identically distributed random variables on $ \mathbb{R} $ with a strictly increasing continuous distribution function F, and the expectation of ∆1 does not exist.

COHOMOLOGY RING OF THE BRST OPERATOR ASSOCIATED TO THE SUM OF TWO PURE SPINORS

In the study of the Type II superstring, it is useful to consider the BRST complex associated to the sum of two pure spinors. The cohomology of this complex is an infinite-dimensional vector space. It is also a finite-dimensional algebra over the algebra of functions of a single pure spinor. In this paper we study the multiplicative structure.

A negative answer to the question of the linearity of Tate’s Trace for the sum of two endomorphisms

The aim of this note is to solve a problem proposed by J. Tate in 1968 by offering a counter-example of the linearity of the trace for the sum of two finite potent operators on an infinite-dimensional vector space.

Determinants of finite potent endomorphisms, symbols and reciprocity laws

The aim of this paper is to offer an algebraic definition of infinite determinants of finite potent endomorphisms using linear algebra techniques. It generalizes Grothendieck’s determinant for finite rank endomorphisms and is equivalent to the classic analytic definitions. The theory can be interpreted as a multiplicative analogue to Tate’s formalism of abstract residues in terms of traces of finite potent linear operators on infinite-dimensional vector spaces, and allows us to relate Tate’s theory to the Segal–Wilson pairing in the context of loop groups.

Perturbation theory for normal operators

Let $E \ni x\mapsto A(x)$ be a $\mathscr {C}$-mapping with its values being unbounded normal operators with common domain of definition and compact resolvent. Here $\mathscr {C}$ stands for $C^\infty$, $C^\omega$ (real analytic), $C^{[M]}$ (Denjoy–Carleman of Beurling or Roumieu type), $C^{0,1}$ (locally Lipschitz), or $C^{k,\alpha }$. The parameter domain $E$ is either $\mathbb R$ or $\mathbb R^n$ or an infinite dimensional convenient vector space. We completely describe the $\mathscr {C}$-dependence on $x$ of the eigenvalues and the eigenvectors of $A(x)$. Thereby we extend previously known results for self-adjoint operators to normal operators, partly improve them, and show that they are best possible. For normal matrices $A(x)$ we obtain partly stronger results.

Weak One-Basedness

We study the notion of weak one-basedness introduced in recent work of Berenstein and Vassiliev. Our main results are that this notion characterizes linearity in the setting of geometric þ-rank 1structures and that lovely pairs of weakly one-based geometric þ-rank 1 structures are weakly one-based with respect to þ-independence. We also study geometries arising from infinite-dimensional vector spaces over division rings.